Gullstrand–Painlevé coordinates

Gullstrand–Painlevé coordinates are a particular set of coordinates for the Schwarzschild metric – a solution to the Einstein field equations which describes a black hole. The ingoing coordinates are such that the time coordinate follows the proper time of a free-falling observer who starts from far away at zero velocity, and the spatial slices are flat. There is no coordinate singularity at the Schwarzschild radius (event horizon). The outgoing ones are simply the time reverse of ingoing coordinates (the time is the proper time along outgoing particles that reach infinity with zero velocity).

The solution was proposed independently by Paul Painlevé in 1921 and Allvar Gullstrand in 1922. It was not explicitly shown until 1933 in Lemaître's paper that these solutions were simply coordinate transformations of the usual Schwarzschild solution, although Einstein immediately believed that to be true.